1 an envelope detection method based on the first vibration mode of bearing vibration

An envelope detection method based on the first-vibration-mode of bearing vibration Department of Mechanical Engineering, Southern Taiwan University, 1 Nan-Tai Street, Yung Kang City, 710

An envelope detection method based on the first-vibration-mode of bearing vibration

Department of Mechanical Engineering, Southern Taiwan University, 1 Nan-Tai Street,

Yung Kang City, 710 Tainan County, Taiwan, ROC Received 29 March 2007; received in revised form 12 September 2007; accepted 23 November 2007

Available online 4 December 2007

Abstract

In this paper, the resonance frequency in the first-vibration-mode of mechanical system is studied and applied in the envelope detection for the bearing vibration The vibration signal of a bearing system is a typical vibration with amplitude modulation Under the assumption of a stepwise function for the envelope signal, the modulated signal could be decom-posed into a sinusoidal function basis at the first-vibration-mode resonance frequency According to the vibration spec-trum, the first-vibration-mode resonance frequency could be initially designated By applying a recursive estimation algorithm, the resonance frequency could be derived more precisely Thus, the envelope signal could be retrieved by esti-mating the coefficients of the function basis with the linear least squares analysis In addition, the vibration signal with noise rejection could be directly reconstructed from the envelope signal According to the experimental study, the envelope detection method for the first-vibration-mode resonance frequency could be effectively applied in the signal processing for the bearing defect diagnosis

Ó 2007 Elsevier Ltd All rights reserved

Keywords: Resonance frequency; Amplitude modulation; Envelope detection; Bearing defect

1 Introduction

The vibration signal of a bearing system is a

typ-ical vibration with amplitude modulation The

high-frequency resonance technique[1]is usually applied

to the mechanical vibrations with amplitude

modu-lation When applying the high-frequency resonance

technique to detect the envelope signal, the

vibra-tion signal is operated through a bandpass filter to derive a single mode vibration and then the envelop-ing transformation is applied to retrieve an envelope signal In the range of the system resonance, this technique takes advantage of the absence of low-fre-quency mechanical noise to demodulate a vibration signal and, therefore, provides a low-frequency envelope signal with a high signal-to-noise ratio

In order to implement the high-frequency resonance technique, the Hilbert transform is often applied in vibration signal demodulation to provide a complex signal Accordingly, the envelope signal could be

0263-2241/$ - see front matter Ó 2007 Elsevier Ltd All rights reserved.

doi:10.1016/j.measurement.2007.11.007

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Measurement 41 (2008) 797–809

www.elsevier.com/locate/measurement

obtained from the absolute value of the complex

method in the envelope detection of vibration

signals

Although the high-frequency resonance

tech-nique is widely applied in the envelope detection

of bearing vibration with amplitude modulation, it

is difficult to properly designate the bandpass filter

through which a complete mode vibration could

be filtered There is no guarantee a complete mode

vibration of bearing vibrations could be filtered

through such a bandpass filter Thus, it could

seri-ously distort the envelope signal In addition, when

applying the Hilbert transform in the

high-fre-quency resonance technique for the envelope

detec-tion of vibradetec-tion signals, the computing burden

would be very high

In this paper, a signal processing method and a

recursive estimation algorithm are proposed to

apply in the vibration signal with amplitude

modulation Based on the resonance frequency in

the first-vibration-mode of a mechanical system,

the envelope detection for the vibration signal with

amplitude modulation could be easily and fast

achieved by the linear least squares analysis

According to the vibration spectrum, the resonance

frequency in the first-vibration-mode would be

ini-tially designated A recursive estimation algorithm

would be then proposed to improve the precision

of estimation for the resonance frequency Under

the assumption of a stepwise function for the

enve-lope signal, the vibration signal could be

decom-posed into a sinusoidal function basis at the

resonance frequency Thus, the envelope signal

could be directly retrieved by estimating the

coeffi-cients of the function basis with the linear least

squares analysis Besides, the vibration signal with

noise rejection could be easily reconstructed from

the envelope signal In the experimental study, the

effectiveness of the envelope detection method with

the first-vibration-mode resonance frequency would

be investigated and applied in the bearing defect diagnosis

2 A study on the bearing vibrations 2.1 Amplitude modulation of bearing vibration

In a bearing system, the carrier signal could be a combination of the resonant frequencies of the bear-ing or even of the mechanical system, and thus the vibration signal with amplitude modulation could

be represented as [6–8]

where w(t) is the low-frequency mechanical noise, and the first two summation terms describe the vibrations of defect components and normal compo-nents In the first term of Eq.(1), md is the number of defect, dm(t) depicts the defect impulse train of con-tact and is the modulating signal, qm(t) describes the dimension information of defect and the sensitivity

of striking energy, and alm(t) is the characteristic function of transmission path In the other term, nr

is the number of roller, gn(t) describes the surface functions of normal bearing components with respect to roller n and could also be a modulating signal, qn(t) is the equivalent stiffness of roller n and is a function of the structure stiffness and the oil film stiffness, and aln(t) is the transmission path function which describes the vibration strength excited by roller n For the other variables, rl and

fl are respectively the exponential damping fre-quency and the carrier frefre-quency, and would be the intrinsic characteristics of the system L is the quantity of vibration mode of the system hlm(t) and hln(t) are the initial angles for the amplitude modulation It should be noted that the frequencies

of modulating signal dm(t) and gn(t) would be always much smaller than that of the carrier signal, and the frequencies of modulating signal dm(t) and gn(t) are higher than those of qm(t), alm(t), qn(t) and aln(t)

expanded in frequency band whose center frequency

vðtÞ ¼XL

l¼1

Xmd

m¼1

Z t

1

þXnr

n¼1

Z t

1

!

would be at the frequencies of carrier signal This

phenomenon is named as amplitude modulation

Suppose that the impulse responses due to

impacts dm(t) are completely died out in a time

interval between two consecutive impact contacts,

and the resonance frequency flis high The impact

energy due to the surface irregularity gn(t)qn(t)

would be much smaller than that due to the bearing

defects Thus, the second term of Eq (1) could be

neglected Accordingly, the envelope signal of the

lth mode vibration could be written as

elðtÞ ¼Xmd

m¼1

with umðtÞ ¼ er l t0 and t0= mod(t,1/fdm), where wl(t)

is the low-frequency mechanical noise occurring in

thelth vibration mode, mod(t,1/fdm) represents a

residue of t, and fdmis the frequency of impulse train

dm(t) The spectrum for the envelope signal would

show a pattern of the modulating signal frequency

and its harmonics with equal frequency spacing

sidebands which are induced by the frequencies of

qm(t) and alm(t) When applying the high-frequency

resonance technique to detect the envelope signal,

the vibration signal of defect bearing is operated

through a bandpass filter to derive a single vibration

mode and then taking the enveloping

transforma-tion to retrieve an envelope signal However, there

is no guarantee such a signal processing could

de-rived a complete mode vibration Thus, it could

seri-ously distort the envelope signal

2.2 Envelope detection for bearing vibrations

In general, the resonant vibration modes of a

mechanical system would be more than one and

the structure should be strong enough to stably

sup-port the bearing system Thus, the

first-vibration-mode resonance frequency could be high and away

from the range of low-frequency noise In this paper, the first mode vibration is suggested to be fil-tered out in order to reduce the computing burden According to Eq (1), the vibration signal decom-posed into a sinusoidal function basis at the reso-nance frequency f1 in the first-vibration-mode for

a defect bearing could be represented as

v1ðtÞ ¼Xmd m¼1

umðtÞqmðtÞa1mðtÞ cosð2pf1tþ h1mÞ þ w1ðtÞ

¼ aðtÞ cosð2pf1tÞ þ bðtÞ sinð2pf1tÞ þ w1ðtÞ ð3Þ where a(t) and b(t) are the coefficients for the vibra-tion signal v1(t) mapping to a sinusoidal function basis It is noted that the envelope signal would be-come smoother, and the envelope signal could thus

be approximated by a stepwise function Moreover,

it is assumed that every time period of the carrier signal could be divided into two intervals for the envelope detection Accordingly, the above signal could be expressed in a discrete mode

v1ðiÞ ¼ ajcos 2pf1

i

fs

þ bjsin 2pf1

i

fs

þ wj; ð4Þ where i, fsand wjare the sampling point, the sam-pling frequency of the vibration signal, and the vibration noise induced by the vibration noise

wl(t), respectively j is the interval number of the envelope signal Accordingly, the sampling fre-quency fs must be at least 6 times higher than the resonance frequency f1 In addition, an anti-aliasing low-pass filter with the cutoff frequency between f1 and 0.5fsHz should be adopted

As shown in Fig 1, there are more than three equations to solve the unknown coefficients aj, bj and wj If the number of data points is h over the kth half period of the vibration signal v1(t), these points could be expressed in the matrix form

v1ðhðk  1Þ þ 1Þ

v1ðhðk  1Þ þ 2Þ

vlðhkÞ

2

6

6

4

3 7 7

cos 2pf1 ðk1Þþ1

sin 2pf1 ðk1Þþ1

1 cos 2pf1 ðk1Þþ2

sin 2pf1 ðk1Þþ2

1

cos 2pf1hkfs

sin 2pf1hkfs

1

2 6 6 6 6

3 7 7 7 7

ak

bk

wk

2 6 3

The above equation could be rewritten as a

simpli-fied expression

½Vkh1¼ ½Mkh3

ak

bk

wk

2 6

3

The solution of the matrix function in three

un-knowns could be obtained in a linear least squares

sense by simple equations

ak

bk

wk

2

6

3

7

5 ¼ ½Mk1h3½Vkh1; for h¼ 3

ak

bk

wk

2

6

3

7

5 ¼ ½M kTh3½Mkh31

½MkTh3½Vkh1; for h > 3

8

>

>

>

>

>

>

>

>

ð7Þ

where T denote the transpose of a matrix

Accord-ingly, the envelope signal in Eq.(2)could be derived

from the following equation:

e1ðjÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a2

j

q

Moreover, based on the envelope signal, a

recon-structed vibration signal with noise rejection could

be directly derived from

v01ðiÞ ¼ ajcos 2pf1

i f

þ bjsin 2pf1

i f

Accordingly, Eq (9) is the vibration signal corre-sponding to the first mode vibration for the vibra-tion signal

2.3 The designation of the first-vibration-mode resonance frequency for bearing vibrations However, the most important problem in the implementation of the above envelope detection method is how to designate the resonance frequency

in the range of the first-vibration-mode for a vibra-tion signal In general, an easy and feasible way to designate the resonance frequency is to choose a peak

at the frequency ^f1in the range of the first-vibration-mode Thus, the peak frequency ^f1would be an esti-mation of the resonance frequency in the range of the first mode vibration It is very possible that an error

fre-quency Thus, the actual resonance frequency would

be f1¼ ^f1þ Df The vibration signal v0

1ðiÞ of first-vibration-mode decomposed into a sinusoidal func-tion basis at the estimated resonance frequency could

be an estimation of Eq.(9)and would be written as

v01ðiÞ ¼ a0jcos 2p^f1

i

fs

þ b0jsin 2p^f1

i

fs

¼ a0

jcos 2pðf1 Df Þ i

fs

þ b0j

 sin 2pðf1 Df Þ i

f

ð10Þ

Time

v l (k 1 )=α1 cos(2πf l k 1 /f s )+β1 sin(2pi f l k 1 /f s ), k 1 =1, 2, ,5

v l (k 1 )=α2 cos(2πf l k 1 /f s )+β2 sin(2πf l k 1 /f s ), k 1 =6, 7, ,10

v

l (k

1 )=α3 cos(2πf

l k

1 /f

s )+β3 sin(2πf

l k

1 /f

s ), k

1 =11, 12, ,15

v

l (k

1 )=α4 cos(2πf

l k

1 /f

s )+β4 sin(2πf

l k

1 /f

s ), k

1 =16, 17, ,20

v l (k 1 )=α3 cos(2πf l k 1 /f s )+β3 sin(2πf l k 1 /f s ), k 1 =21, 22, ,25

Fig 1 The modulated signal for a vibration impact decomposes into a sinusoidal function at its resonant frequency.

The above equation could be also rewritten as a

sinusoidal function with fundamental frequency f1,

and be expressed as

v01ðiÞ ¼ a0jcos 2pDf i

fs

 b0jsin 2pDf i

fs

i

fs

þ b0jcos 2pDf i

fs



þa0

jsin 2pDf i

fs

sin 2pf1

i

fs

ð11Þ

In comparison between Eqs.(9) and (11), if v0

1ðiÞ ¼

v1ðiÞ the following equations could be derived:

aj¼ a0

fs

 b0jsin 2pDf i

fs

ð12:aÞ

bj¼ b0jcos 2pDf i

fs

þ a0jsin 2pDf i

fs

ð12:bÞ

If substituting Eq (12) into Eq (8), the envelope

signal could also be written as

e1ðjÞ ¼ ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi

a02

q

It could prove that the envelope signal could be

accurately retrieved, even though, by applying an

estimation of the resonance frequency Thus, an

estimation of the resonance frequency f1 could be

easily derived from the peak in the

first-vibration-mode of the spectrum for the vibration signal Thus,

the proposed envelope detection method would be

capable of applying in practice

Nevertheless, the estimation of the resonance

fre-quency should be accurate enough to apply in Eq.(4)

to derive the envelope signal In the following, a

recursive estimation algorithm for deriving the

first-vibration-mode resonance frequency is proposed:

1 According to the vibration spectrum of the

mea-sured signal, a peak in the first vibration mode

would be initially designated as the resonance

frequency

2 Applying the resonance frequency, the envelope

signal could be derived from the linear least

square analysis, as shown in Eqs (7) and (8)

3 The vibration signal with noise rejection could be

reconstructed from the envelope signal, as shown

in Eq (9)

4 According to the vibration spectrum of the

reconstructed signal derived in Step 3, the peak

would be designated as the resonance frequency

5 Repeat Steps 2–4, until the resonance frequency

is converged

Theoretically, a resonance frequency must exist within the range of first vibration mode on the vibration spectrum The above algorithm should not diverge in estimating the resonance frequency

3 Experimental study

In the following, the applications of the proposed method on the vibration signals of tapered roller bearings (SKF type 32208) are studied The electri-cal-discharge machining method is applied to pro-duce artificial defect on the surface of bearing components which are roller, outer race and inner race The defect sizes are described in theTable 1 The vibration signals are measured on the housing

of the test bearing The measured direction is radial

to the shaft

Under different running speeds, the proposed envelope detection method is applied and the results are shown inFigs 2–4, respectively According to Figs 2a, 3a and 4a, the passband frequencies are similar under different running speeds except that the amplitude is increased, especially at the high-fre-quency band, with the running speed There could

be four resonance modes with the passband fre-quencies from 1 to 3 kHz, 3 to 5 kHz, 6 to 8 kHz and 8 to 10 kHz, respectively In addition, it is noted that the four resonance modes are also similar under different types of bearing defect Thus, the cutoff frequency of the low-pass filter for deriving the first-vibration-mode of bearing vibrations could be designated to be 3 kHz and a sampling rate 18

4b, the initial values designated for the first-vibra-tion-mod resonance frequencies are 2386, 2487 and 1671 Hz, respectively By applying the recursive estimation method in the estimation of the reso-nance frequencies, the vibration spectra are shown

in Figs 2c, 3c and 4c with at the first-vibration-mod resonance frequencies at 2386, 2487 and

2519 Hz, respectively It is noted that the resonance

Table 1 Defect sizes of defective bearings Defect type Defect size (length  width  depth) Roller defect 16 mm  0.15 mm  0.1 mm Outer-race defect 14 mm  0.15 mm  0.1 mm Inner-race defect 18.5 mm  0.15 mm  0.1 mm

recursive iterations In addition, the vibration

spec-tra shown inFigs 2c, 3c and 4cfor the reconstructed

signals are almost the same as those shown inFigs

2b, 3b and 4b, respectively, in the passband from

1.5 to 3 kHz for the measured signals

From the theoretical study, it is assumed that the first mode vibration should be filtered through an anti-aliasing low-pass filter According to the above experimental study, it would prove that the first vibration mode of vibration signal could be filtered

0 0.1 0.2 0.3

Frequency (Hz)

a

1st vibration mode

2nd vibration mode

3rd vibration mode

4th vibration mode

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

0.1 0.2

*

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

0.1 0.2

Frequency (Hz)

* :the initial designated resonance frequency b

c

* :the first-vibration-mode resonance frequency

*

Fig 2 The vibration spectra for the roller defect bearing running at 800 rpm (a) For the vibration signal with a passband from 0 to

12 kHz, (b) for the vibration signal with a passband from 0 to 3 kHz, (c) for the reconstructed signal derived from the envelope signal.

through an anti-aliasing low-pass filter with the

cut-off frequency at 3 kHz Thus, the characteristics of

the vibration signal in the first vibration mode could

be very well represented by the reconstructed signal

in Eq (9)

In the following,Fig 5shows the effectiveness of the proposed method on the envelope detection for

a roller defect bearing at the running speed

1600 rpm Fig 5a–c shows the measured vibration signal, the retrieved envelope signal from Fig 5a,

0 1 2 3 4 5

Frequency (Hz)

a

1st vibration mode

2nd vibration mode

3rd vibration mode

4th vibration mode

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

0.5 1 1.5

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

0.5 1 1.5

Frequency (Hz)

* :the initial designated resonance frequency

*

b

* :the first-vibration-mode resonance frequency

*

c

Fig 3 The vibration spectra for the roller defect bearing running at 1600 rpm (a) For the vibration signal with a passband from 0 to

12 kHz, (b) for the vibration signal with a passband from 0 to 3 kHz, (c) for the reconstructed signal derived from the envelope signal.

and the reconstructed signal derived fromFig 5b,

respectively For the purpose of comparison, the

envelope signal derived from the high-frequency

resonant technique with the passband from 1 to

Fig 5d,Fig 5b shows a smoother signal with lower noise In addition,Fig 6a and b show the envelope spectra for the envelope signal in Fig 5b and d, respectively The envelope spectra are also similar

0 3 6 9

Frequency (Hz)

a

1st vibration mode

2nd vibration mode

4th vibration mode

3rd vibration mode

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

1 2 3

*

0 500 1000 1500 2000 2500 3000 3500 4000 4500 5000 0

1 2 3

Frequency (Hz)

* :the initial designated resonance frequency

b

* :the first-vibration-mode resonance frequency c

*

Fig 4 The vibration spectra for the roller defect bearing running at 2400 rpm (a) For the vibration signal with a passband from 0 to

12 kHz, (b) for the vibration signal with a passband from 0 to 3 kHz, (c) for the reconstructed signal derived from the envelope signal.

frequency at 154 Hz for the roller defect Thus, it

would prove that the proposed method could be

effective in the envelope detection for the bearing vibration

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -2

0 2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0

0.5 1

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 -2

0 2

0 0.01 0.02 0.03 0.04 0.05 0.06 0.07 0.08 0.09 0.1 0

0.5 1

Time (sec)

a

b

c

d

Fig 5 (a) The measured vibration signal for the roller defect bearing running at 1600 rpm, (b) the envelope signal of (a) in the first vibration mode, (c) the reconstructed signal from (b), (d) the envelope signal derived from the high-frequency resonance technique.

0 2 4 6 8

0 3 6 9 12

Frequency (Hz)

a

b

Fig 6 The envelope spectra for a roller defect bearing running at 1600 rpm (a) By applying the envelope detection with the first-vibration-mode resonance frequency, (b) by applying the high frequency resonance technique with a filtering passband from 1 to 3 kHz.

Similar to Figs 6–8 show the envelope spectra

with the characteristic frequency at 190 Hz and

266 Hz for the outer-race defect bearing and the inner-race defect bearing, respectively It is found

0 2 4 6 8 10

0 2 4 6 8 10

Frequency (Hz)

a

b

Fig 7 The envelope spectra for a outer-race defect bearing running at 1600 rpm (a) By applying the envelope detection with the first-vibration-mode resonance frequency (b) By applying the high frequency resonance technique with a filtering passband from 1 to 3 kHz.

0 1 2

0 1 2

Frequency (Hz)

a

b

Fig 8 The envelope spectra for a inner-race defect bearing running at 1600 rpm (a) By applying the envelope detection with the first-vibration-mode resonance frequency (b) By applying the high frequency resonance technique with a filtering passband from 1 to 3 kHz.